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Closure operators and concept equations in non-commutative fuzzy logic. (English) Zbl 1065.03016

Fuzzy sets for a noncommutative framework are studied in this paper, namely, functions from a set \(X\) to a generalized residuated lattice \(L\). The latter are defined similarly to residuated lattices, but their monoidal operation is no longer assumed to be commutative.
In analogy to the commutative case, \(L\)-closure operators on \(L^X\) and \(L\)-closure systems are defined and their equivalence shown. Furthermore, the notion of a Galois connection between sets \(X\) and \(Y\) is introduced, and it is shown how it may be characterized on the base of left- and right-\(L\)-closure systems on \(L^X\) and \(L^Y\). Finally, \(L\)-Galois connections induced by some binary \(L\)-relation on \(X \times Y\) are considered, and also how this relation may be recovered if the Galois connection is known only partly.

MSC:

03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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